Timeline for Open subset in the flat topology on Spec(R)
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 12, 2013 at 23:02 | comment | added | Matt | Also, for the record, that second comment was a reply to a now deleted comment. I wasn't repeating myself unprovoked. | |
| Feb 12, 2013 at 16:57 | comment | added | Martin Brandenburg | @Matt: You are right. These points are not more general, but more generic. Also EGA talks about générisations (Ch. 0, 2.1.1). Somehow I always read it the wrong way, and probably I'm not the only one? There are several sources using "generalization", for example the Stacks Project stacks.math.columbia.edu/tag/0060 and many research papers. | |
| Feb 12, 2013 at 3:31 | comment | added | Matt | I should probably stop before I upset some people, but seriously, the term is "generization." Look in Hartshorne or any textbook that talks about Zariski spaces or sober spaces or even PlanetMath: planetmath.org/encyclopedia/Generization.html | |
| Feb 12, 2013 at 2:11 | comment | added | Matt | Two people have said this now, so it is making me nervous, but isn't the term "stable under generization"? | |
| Feb 11, 2013 at 22:31 | comment | added | Keenan Kidwell | I'm not sure if it's true in the non-Noetherian case, but if $R$ is Noetherian, then a subset $S$ of $\mathrm{Spec}(R)$ is (Zariski) open if and only if it is constructible (in the sense of the Stacks Project) and stable under generalization. | |
| Feb 11, 2013 at 20:29 | comment | added | Georg S. | @Laurent: Oh, that's a good point. Thank you! | |
| Feb 11, 2013 at 20:25 | comment | added | Laurent Moret-Bailly | The only Zariski neighborhood of the closed point in a local scheme $X$ is $X$. So, $U=\mathrm{Spec\,}(R_P)$ and your condition just means that $S$ is stable under generalization. | |
| Feb 11, 2013 at 19:47 | comment | added | Georg S. | A delicate aspect. I know that as a Grothendieck topology I would have to consider jointly surjective flat morphisms (+ some properties perhaps). But in "Going-down implies generalized going-down" by Dobbs--Hetzel (Lemma 2.1) there is a topology (in the "classical" sense) on Spec(R) defined (but this should have been considered before), which I thought is also referred to as the "flat topology". I clearly don't know how this is connected to the flat topology in the Grothendieck sense. | |
| Feb 11, 2013 at 18:13 | comment | added | Zhen Lin | The flat "topology" is not a topology in the classical sense... | |
| Feb 11, 2013 at 17:50 | history | asked | Georg S. | CC BY-SA 3.0 |